Such a reflector lamp is known from WO-02/48609, in which the support body provides an accurate positioning of the lamp vessel in the reflector. It is a disadvantage of the known reflector lamp that the support body is of a relatively complex construction and that assembly of the lamp is relatively cumbersome. Relatively high stresses are imposed on the (glass) reflector body during said assembly and in an assembled configuration. It is in particular known that high stresses are imposed on the reflector adjacent the opening/neck of the reflector. These stresses are caused by both the mechanical construction and the thermal load on the reflector lamp during operation.